Application of the Constrained Formulation to the Nonlinear Sloshing Problem Based On the ALE Method

Abstract This study deals with an application of constrained formulation to a nonlinear sloshing problem based on the Arbitrary Lagrangian-Eulerian finite element method (ALE). The ALE method incorporates a discretized form of equations of motion with mesh updating algorithms in order to prevent a problem of mesh distortion. This paper focuses on an analytical aspect of such treatments as constrained systems in the formulation of the ALE method. Since the mesh updating algorithms give algebraic relations for nodal coordinates, this study treats these relations as constraints. Then, we introduce formulation for constrained systems based on the method of Lagrange multipliers. As a result of this formulation, equations of motion are given by differential algebraic equations (DAEs) consisting of differential equations for time evolution of physical quantities and algebraic equations (constraints). The present method can be classified into a kind of augmented formulation. In particular, the present approach is motivated by the inherent simplicity of the DAEs. Moreover, we present a matrix size reduction technique used in the Newton-Raphson method in order to remove a part of the redundant degrees of freedom in the iterative procedures, because the resulting set of DAEs involves a larger number of unknowns than the minimal number of degrees of freedom due to the introduction of the constrained formulation. In addition, this study presents a method to introduce damping effects defined in the modal space into the FEM models. The proposed approach is validated by comparisons with experimental data in the time domain analysis.